We discuss an effective method of computing traces, determinants, and $\zeta$-functions for some classes of linear operators and apply this to the concrete case of Sturm-Liouville operators.

To illustrate the formalism, we will sketch a spectral theorist’s computation of $\pi$, Jacobi’s classical transformation formula for one-dimensional theta functions (utilizing the heat equation on the circle), and sketch a derivation of a formula for Ap\’ery’s constant, $\zeta(3)$, employing a trace formula.

The talk will minimize technicalities and be accessible to students.

This is based in part on recent joint work with Klaus Kirsten, Lance Littlejohn, and Hagop Tossounian.